Park et al. [1]} approximate the integral in (REF ) as \(Vp(\textbf {x}_k,\textbf {s})\) , where \(V\)  is the 2D footprint or area occupied by the robot. For computing \(p(\textbf {x}_k,\textbf {s})\) , they first assume a distribution centered around the obstacle with the covariance being the sum of the robot and obstacle location uncertainties. The collision probability is then computed by finding the \(\textbf {x}_k\)  that maximizes \(p(\textbf {x}_k,\textbf {s})\)  and formulate the problem as an optimization problem with a Lagrange multiplier. In [2]} the density of the center of the robot is used. For comparing with these approaches, we formulate the problem as given in each of these worksFor the comparison, the approaches in [2]}, [1]} have been reproduced to the best our understanding and the reproduced codes (including numerical integration and our approach) can be found here— https://bitbucket.org/1729antony/comparison_cp_methods/src/master/. In order to validate the values computed using our approach, we perform numerical integration of the expression in (REF ), which gives the exact collision probability value.
